Second-order Boltzmann equation: gauge dependence and gauge invariance

Atsushi Naruko, Cyril Pitrou, Kazuya Koyama, Misao Sasaki

Research output: Contribution to journalArticlepeer-review


In the context of cosmological perturbation theory, we derive the second-order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: (i) the polarization of light is incorporated in this formalism by using a tensor-valued distribution function; (ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; (iii) we perform a separation between temperature and spectral distortion, both for the intensity and polarization for the first time; (iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature.
Original languageEnglish
Article number165008
Pages (from-to)165008
JournalClassical and Quantum Gravity
Issue number16
Early online date19 Jul 2013
Publication statusPublished - 21 Aug 2013


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